Paradox of the arrow

the paradox of the arrow is a paradox formulated by Zénon d' Élée.

We At every moment imagine a arrow in vol., the arrow is with a precise position. If the moment is too short, then the arrow does not have time to move and remains at rest during this moment. Now, during the following moments, it will remain motionless for the same reason. The arrow is always motionless and cannot move: the movement is impossible.

This Paradoxe is solved mathematically as follows: since the speed of the arrow is not null, the limit of the rate of variation in one moment is not null and thus the rate of variation between two very short moments will not be null. In other words, even if the moment is very short, the arrow will traverse a certain distance.

There exists also a physical solution with this Paradoxe: after all, according to the same reasoning, an object initially at rest could never start, since motionless! Actually, the capacity of an object to be moved at one moment T is not related to the fact that it is mobile or not at this moment T , but with its kinetic energy at this moment. An object “motionless”, but equipped with a certain kinetic energy, will move at the next moment. Now, like, in translation, the kinetic energy is proportional to the square speed, the solutions physics and mathematics is not so distant one from the other which it could appear it with the first glance…

In addition, in the formulation of the Paradox above, there is confusion between moment and moment. Admittedly at a given moment, the arrow is motionless, but one moment is never too short so that an arrow has time to move! If the arrow can be regarded as “motionless” at a given moment, on the other hand, between two successive moments, separated by one even infinitesimal moment, it moves.

Lastly, if one goes at the bottom of the things, even at a given moment, the arrow is not really “motionless”: any photographer knows that an object moving appears on a photograph with a certain blur in the direction of the movement, which distinguishes it from the truly motionless objects. In the same way, mechanics quantum says us that object moving presents certain uncertainty on its position (certainly, this uncertainty is infinitesimal for an object of a also large size and a speed as low as those of an arrow, but it does not exist about it less), which radically distinguishes it from an object at rest, and allows the continuation of the movement.

Notice

It would be naive to believe that Zénon disputed that an arrow can strike a tree. The purpose of the paradoxes which it used were of course to clarify dark zones in the process of certain types of reasoning utilizing the infinite one. It did not call into question the bottom, but the tool which is used to us to think the world, i.e. our brain. The principal difficulty highlighted by its paradoxes comes owing to the fact that time and the movement are concepts essentially continuous, which are not let apprehend in an adequate way by the means of a sequencing. To cut out time without precaution, like one cuts out a cake, leads to nonsenses. The spirit has evil to reason on time, the flow, continuity, the movement or the infinite one. Our intuitions on these subjects are often faulty. That comes from what at the base, the human brain functions by associations of ideas, not by logical deduction. Thus, we are not naturally rational, and we have a preconceived design of continuous for example, which comes to interfere with any reasoning on this subject. In conclusion, we must show ourselves particularly careful when we mix to handle of such concepts.

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